Partial Differential Equations part 7

standard tridiagonal algorithm. Given un , one solves (19.5.36) for un+1/2 , substitutes on the right-hand side of (19.5.37), and then solves for un+1 . The key question is how to choose the iteration parameter r

Nội dung Text: Partial Differential Equations part 7

19.6 Multigrid Methods for Boundary Value Problems 871
standard tridiagonal algorithm. Given un , one solves (19.5.36) for un+1/2 , substitutes
on the right-hand side of (19.5.37), and then solves for un+1 . The key question
is how to choose the iteration parameter r, the analog of a choice of timestep for
an initial value problem.
As usual, the goal is to minimize the spectral radius of the iteration matrix.
Although it is beyond our scope to go into details here, it turns out that, for the
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optimal choice of r, the ADI method has the same rate of convergence as SOR.
The individual iteration steps in the ADI method are much more complicated than
in SOR, so the ADI method would appear to be inferior. This is in fact true if we
choose the same parameter r for every iteration step. However, it is possible to
choose a different r for each step. If this is done optimally, then ADI is generally
more efﬁcient than SOR. We refer you to the literature [1-4] for details.
Our reason for not fully implementing ADI here is that, in most applications,
it has been superseded by the multigrid methods described in the next section. Our
advice is to use SOR for trivial problems (e.g., 20 × 20), or for solving a larger
problem once only, where ease of programming outweighs expense of computer
time. Occasionally, the sparse matrix methods of §2.7 are useful for solving a set
of difference equations directly. For production solution of large elliptic problems,
however, multigrid is now almost always the method of choice.
CITED REFERENCES AND FURTHER READING:
Hockney, R.W., and Eastwood, J.W. 1981, Computer Simulation Using Particles (New York:
McGraw-Hill), Chapter 6.
Young, D.M. 1971, Iterative Solution of Large Linear Systems (New York: Academic Press). [1]
Stoer, J., and Bulirsch, R. 1980, Introduction to Numerical Analysis (New York: Springer-Verlag),
§§8.3–8.6. [2]
Varga, R.S. 1962, Matrix Iterative Analysis (Englewood Cliffs, NJ: Prentice-Hall). [3]
Spanier, J. 1967, in Mathematical Methods for Digital Computers, Volume 2 (New York: Wiley),
Chapter 11. [4]
19.6 Multigrid Methods for Boundary Value
Problems
Practical multigrid methods were ﬁrst introduced in the 1970s by Brandt. These
methods can solve elliptic PDEs discretized on N grid points in O(N ) operations.
The “rapid” direct elliptic solvers discussed in §19.4 solve special kinds of elliptic
equations in O(N log N ) operations. The numerical coefﬁcients in these estimates
are such that multigrid methods are comparable to the rapid methods in execution
speed. Unlike the rapid methods, however, the multigrid methods can solve general
elliptic equations with nonconstant coefﬁcients with hardly any loss in efﬁciency.
Even nonlinear equations can be solved with comparable speed.
Unfortunately there is not a single multigrid algorithm that solves all elliptic
problems. Rather there is a multigrid technique that provides the framework for
solving these problems. You have to adjust the various components of the algorithm
within this framework to solve your speciﬁc problem. We can only give a brief

872 Chapter 19. Partial Differential Equations
introduction to the subject here. In particular, we will give two sample multigrid
routines, one linear and one nonlinear. By following these prototypes and by
perusing the references [1-4] , you should be able to develop routines to solve your
own problems.
There are two related, but distinct, approaches to the use of multigrid techniques.
The ﬁrst, termed “the multigrid method,” is a means for speeding up the convergence
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of a traditional relaxation method, as deﬁned by you on a grid of pre-speciﬁed
ﬁneness. In this case, you need deﬁne your problem (e.g., evaluate its source terms)
only on this grid. Other, coarser, grids deﬁned by the method can be viewed as
temporary computational adjuncts.
The second approach, termed (perhaps confusingly) “the full multigrid (FMG)
method,” requires you to be able to deﬁne your problem on grids of various sizes
(generally by discretizing the same underlying PDE into different-sized sets of ﬁnite-
difference equations). In this approach, the method obtains successive solutions on
ﬁner and ﬁner grids. You can stop the solution either at a pre-speciﬁed ﬁneness, or
you can monitor the truncation error due to the discretization, quitting only when
it is tolerably small.
In this section we will ﬁrst discuss the “multigrid method,” then use the concepts
developed to introduce the FMG method. The latter algorithm is the one that we
implement in the accompanying programs.
From One-Grid, through Two-Grid, to Multigrid
The key idea of the multigrid method can be understood by considering the
simplest case of a two-grid method. Suppose we are trying to solve the linear
elliptic problem
Lu = f (19.6.1)
where L is some linear elliptic operator and f is the source term. Discretize equation
(19.6.1) on a uniform grid with mesh size h. Write the resulting set of linear
algebraic equations as
Lh uh = fh (19.6.2)
Let uh denote some approximate solution to equation (19.6.2). We will use the
symbol uh to denote the exact solution to the difference equations (19.6.2). Then
the error in uh or the correction is
vh = uh − uh (19.6.3)
The residual or defect is
dh = Lh uh − fh (19.6.4)
(Beware: some authors deﬁne residual as minus the defect, and there is not universal
agreement about which of these two quantities 19.6.4 deﬁnes.) Since Lh is linear,
the error satisﬁes
Lh vh = −dh (19.6.5)

19.6 Multigrid Methods for Boundary Value Problems 873
At this point we need to make an approximation to Lh in order to ﬁnd vh . The
classical iteration methods, such as Jacobi or Gauss-Seidel, do this by ﬁnding, at
each stage, an approximate solution of the equation
Lh vh = −dh (19.6.6)
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where Lh is a “simpler” operator than Lh . For example, Lh is the diagonal part of
Lh for Jacobi iteration, or the lower triangle for Gauss-Seidel iteration. The next
approximation is generated by
unew = uh + vh
h (19.6.7)
Now consider, as an alternative, a completely different type of approximation
for Lh , one in which we “coarsify” rather than “simplify.” That is, we form some
appropriate approximation LH of Lh on a coarser grid with mesh size H (we will
always take H = 2h, but other choices are possible). The residual equation (19.6.5)
is now approximated by
LH vH = −dH (19.6.8)
Since LH has smaller dimension, this equation will be easier to solve than equation
(19.6.5). To deﬁne the defect dH on the coarse grid, we need a restriction operator
R that restricts dh to the coarse grid:
dH = Rdh (19.6.9)
The restriction operator is also called the ﬁne-to-coarse operator or the injection
operator. Once we have a solution vH to equation (19.6.8), we need a prolongation
operator P that prolongates or interpolates the correction to the ﬁne grid:
vh = P vH (19.6.10)
The prolongation operator is also called the coarse-to-ﬁne operator or the inter-
polation operator. Both R and P are chosen to be linear operators. Finally the
approximation uh can be updated:
unew = uh + vh
h (19.6.11)
One step of this coarse-grid correction scheme is thus:
Coarse-Grid Correction
• Compute the defect on the ﬁne grid from (19.6.4).
• Restrict the defect by (19.6.9).
• Solve (19.6.8) exactly on the coarse grid for the correction.
• Interpolate the correction to the ﬁne grid by (19.6.10).

874 Chapter 19. Partial Differential Equations
• Compute the next approximation by (19.6.11).
Let’s contrast the advantages and disadvantages of relaxation and the coarse-grid
correction scheme. Consider the error vh expanded into a discrete Fourier series. Call
the components in the lower half of the frequency spectrum the smooth components
and the high-frequency components the nonsmooth components. We have seen that
relaxation becomes very slowly convergent in the limit h → 0, i.e., when there are a
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large number of mesh points. The reason turns out to be that the smooth components
are only slightly reduced in amplitude on each iteration. However, many relaxation
methods reduce the amplitude of the nonsmooth components by large factors on
each iteration: They are good smoothing operators.
For the two-grid iteration, on the other hand, components of the error with
wavelengths < 2H are not even representable on the coarse grid and so cannot be
∼
reduced to zero on this grid. But it is exactly these high-frequency components that
can be reduced by relaxation on the ﬁne grid! This leads us to combine the ideas
of relaxation and coarse-grid correction:
Two-Grid Iteration
• Pre-smoothing: Compute uh by applying ν1 ≥ 0 steps of a relaxation
¯
method to uh .
• Coarse-grid correction: As above, using uh to give unew .
¯ ¯h
• Post-smoothing: Compute unew by applying ν2 ≥ 0 steps of the relaxation
h
method to unew .
¯h
It is only a short step from the above two-grid method to a multigrid method.
Instead of solving the coarse-grid defect equation (19.6.8) exactly, we can get
an approximate solution of it by introducing an even coarser grid and using the
two-grid iteration method. If the convergence factor of the two-grid method is
small enough, we will need only a few steps of this iteration to get a good enough
approximate solution. We denote the number of such iterations by γ. Obviously
we can apply this idea recursively down to some coarsest grid. There the solution
is found easily, for example by direct matrix inversion or by iterating the relaxation
scheme to convergence.
One iteration of a multigrid method, from ﬁnest grid to coarser grids and back
to ﬁnest grid again, is called a cycle. The exact structure of a cycle depends on
the value of γ, the number of two-grid iterations at each intermediate stage. The
case γ = 1 is called a V-cycle, while γ = 2 is called a W-cycle (see Figure 19.6.1).
These are the most important cases in practice.
Note that once more than two grids are involved, the pre-smoothing steps after
the ﬁrst one on the ﬁnest grid need an initial approximation for the error v. This
should be taken to be zero.
Smoothing, Restriction, and Prolongation Operators
The most popular smoothing method, and the one you should try ﬁrst, is
Gauss-Seidel, since it usually leads to a good convergence rate. If we order the mesh
points from 1 to N , then the Gauss-Seidel scheme is
N
1
ui = − Lij uj − fi i = 1, . . . , N (19.6.12)
j=1
Lii
j=i

19.6 Multigrid Methods for Boundary Value Problems 875
S S
2-grid
E
S S S S
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3-grid
S S S S S
E E E
S S S S
S S 4-grid S S S
S S S S S S S S
E E E E E
γ=1 γ=2
Figure 19.6.1. Structure of multigrid cycles. S denotes smoothing, while E denotes exact solution
on the coarsest grid. Each descending line \ denotes restriction (R) and each ascending line / denotes
prolongation (P ). The ﬁnest grid is at the top level of each diagram. For the V-cycles (γ = 1) the E
step is replaced by one 2-grid iteration each time the number of grid levels is increased by one. For the
W-cycles (γ = 2), each E step gets replaced by two 2-grid iterations.
where new values of u are used on the right-hand side as they become available. The
exact form of the Gauss-Seidel method depends on the ordering chosen for the mesh
points. For typical second-order elliptic equations like our model problem equation
(19.0.3), as differenced in equation (19.0.8), it is usually best to use red-black
ordering, making one pass through the mesh updating the “even” points (like the red
squares of a checkerboard) and another pass updating the “odd” points (the black
squares). When quantities are more strongly coupled along one dimension than
another, one should relax a whole line along that dimension simultaneously. Line
relaxation for nearest-neighbor coupling involves solving a tridiagonal system, and
so is still efﬁcient. Relaxing odd and even lines on successive passes is called zebra
relaxation and is usually preferred over simple line relaxation.
Note that SOR should not be used as a smoothing operator. The overrelaxation
destroys the high-frequency smoothing that is so crucial for the multigrid method.
A succint notation for the prolongation and restriction operators is to give their
symbol. The symbol of P is found by considering vH to be 1 at some mesh point
(x, y), zero elsewhere, and then asking for the values of PvH . The most popular
prolongation operator is simple bilinear interpolation. It gives nonzero values at
the 9 points (x, y), (x + h, y), . . . , (x − h, y − h), where the values are 1, 1 , . . . , 1 .
2 4

876 Chapter 19. Partial Differential Equations
Its symbol is therefore
1 1 1

4 2 4
 1 1 
 2
1 2  (19.6.13)
1 1 1
4 2 4
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The symbol of R is deﬁned by considering vh to be deﬁned everywhere on the
ﬁne grid, and then asking what is Rvh at (x, y) as a linear combination of these
values. The simplest possible choice for R is straight injection, which means simply
ﬁlling each coarse-grid point with the value from the corresponding ﬁne-grid point.
Its symbol is “[1].” However, difﬁculties can arise in practice with this choice. It
turns out that a safe choice for R is to make it the adjoint operator to P. To deﬁne the
adjoint, deﬁne the scalar product of two grid functions uh and vh for mesh size h as
uh |vh h ≡ h2 uh (x, y)vh (x, y) (19.6.14)
x,y
Then the adjoint of P, denoted P † , is deﬁned by
uH |P † vh H = PuH |vh h (19.6.15)
Now take P to be bilinear interpolation, and choose uH = 1 at (x, y), zero elsewhere.
Set P † = R in (19.6.15) and H = 2h. You will ﬁnd that
(Rvh )(x,y) = 1 vh (x, y) + 1 vh (x + h, y) +
4 8
1
16 vh (x + h, y + h) + · · · (19.6.16)
so that the symbol of R is
 1 1 1

16 8 16
 1 1 1 
 8 4 8  (19.6.17)
1 1 1
16 8 16
Note the simple rule: The symbol of R is 1 the transpose of the matrix deﬁning the
4
symbol of P, equation (19.6.13). This rule is general whenever R = P † and H = 2h.
The particular choice of R in (19.6.17) is called full weighting. Another popular
choice for R is half weighting, “halfway” between full weighting and straight
injection. Its symbol is
 
0 1 0
8
1 1 1
8 2 8 (19.6.18)
1
0 8 0
A similar notation can be used to describe the difference operator Lh . For
example, the standard differencing of the model problem, equation (19.0.6), is
represented by the ﬁve-point difference star
 
0 1 0
1 
Lh = 2 1 −4 1  (19.6.19)
h
0 1 0

19.6 Multigrid Methods for Boundary Value Problems 877
If you are confronted with a new problem and you are not sure what P and R
choices are likely to work well, here is a safe rule: Suppose mp is the order of the
interpolation P (i.e., it interpolates polynomials of degree mp − 1 exactly). Suppose
mr is the order of R, and that R is the adjoint of some P (not necessarily the P you
intend to use). Then if m is the order of the differential operator Lh , you should
satisfy the inequality mp + mr > m. For example, bilinear interpolation and its
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adjoint, full weighting, for Poisson’s equation satisfy mp + mr = 4 > m = 2.
Of course the P and R operators should enforce the boundary conditions for
your problem. The easiest way to do this is to rewrite the difference equation to
have homogeneous boundary conditions by modifying the source term if necessary
(cf. §19.4). Enforcing homogeneous boundary conditions simply requires the P
operator to produce zeros at the appropriate boundary points. The corresponding
R is then found by R = P † .
Full Multigrid Algorithm
So far we have described multigrid as an iterative scheme, where one starts
with some initial guess on the ﬁnest grid and carries out enough cycles (V-cycles,
W-cycles,. . . ) to achieve convergence. This is the simplest way to use multigrid:
Simply apply enough cycles until some appropriate convergence criterion is met.
However, efﬁciency can be improved by using the Full Multigrid Algorithm (FMG),
also known as nested iteration.
Instead of starting with an arbitrary approximation on the ﬁnest grid (e.g.,
uh = 0), the ﬁrst approximation is obtained by interpolating from a coarse-grid
solution:
uh = PuH (19.6.20)
The coarse-grid solution itself is found by a similar FMG process from even coarser
grids. At the coarsest level, you start with the exact solution. Rather than proceed as
in Figure 19.6.1, then, FMG gets to its solution by a series of increasingly tall “N’s,”
each taller one probing a ﬁner grid (see Figure 19.6.2).
Note that P in (19.6.20) need not be the same P used in the multigrid cycles.
It should be at least of the same order as the discretization Lh , but sometimes a
higher-order operator leads to greater efﬁciency.
It turns out that you usually need one or at most two multigrid cycles at each
level before proceeding down to the next ﬁner grid. While there is theoretical
guidance on the required number of cycles (e.g., [2]), you can easily determine it
empirically. Fix the ﬁnest level and study the solution values as you increase the
number of cycles per level. The asymptotic value of the solution is the exact solution
of the difference equations. The difference between this exact solution and the
solution for a small number of cycles is the iteration error. Now ﬁx the number of
cycles to be large, and vary the number of levels, i.e., the smallest value of h used. In
this way you can estimate the truncation error for a given h. In your ﬁnal production
code, there is no point in using more cycles than you need to get the iteration error
down to the size of the truncation error.
The simple multigrid iteration (cycle) needs the right-hand side f only at the
ﬁnest level. FMG needs f at all levels. If the boundary conditions are homogeneous,

878 Chapter 19. Partial Differential Equations
S S
S S S S
4-grid
S S S S S ncycle = 1
S
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E E E E
S S S
S S S S S S S
S S S S S S S S S S S 4-grid
ncycle = 2
E E E E E E E
Figure 19.6.2. Structure of cycles for the full multigrid (FMG) method. This method starts on the
coarsest grid, interpolates, and then reﬁnes (by “V’s”), the solution onto grids of increasing ﬁneness.
you can use fH = Rfh . This prescription is not always safe for inhomogeneous
boundary conditions. In that case it is better to discretize f on each coarse grid.
Note that the FMG algorithm produces the solution on all levels. It can therefore
be combined with techniques like Richardson extrapolation.
We now give a routine mglin that implements the Full Multigrid Algorithm
for a linear equation, the model problem (19.0.6). It uses red-black Gauss-Seidel as
the smoothing operator, bilinear interpolation for P, and half-weighting for R. To
change the routine to handle another linear problem, all you need do is modify the
functions relax, resid, and slvsml appropriately. A feature of the routine is the
dynamical allocation of storage for variables deﬁned on the various grids.
#include "nrutil.h"
#define NPRE 1 Number of relaxation sweeps before . . .
#define NPOST 1 . . . and after the coarse-grid correction is com-
#define NGMAX 15 puted.
void mglin(double **u, int n, int ncycle)
Full Multigrid Algorithm for solution of linear elliptic equation, here the model problem (19.0.6).
On input u[1..n][1..n] contains the right-hand side ρ, while on output it returns the solution.
The dimension n must be of the form 2j + 1 for some integer j. (j is actually the number of
grid levels used in the solution, called ng below.) ncycle is the number of V-cycles to be
used at each level.
{
void addint(double **uf, double **uc, double **res, int nf);
void copy(double **aout, double **ain, int n);
void fill0(double **u, int n);
void interp(double **uf, double **uc, int nf);
void relax(double **u, double **rhs, int n);
void resid(double **res, double **u, double **rhs, int n);
void rstrct(double **uc, double **uf, int nc);
void slvsml(double **u, double **rhs);

882 Chapter 19. Partial Differential Equations
• The defect dh vanishes identically at all black mesh points after a red-black
Gauss-Seidel step. Thus dH = Rdh for half-weighting reduces to simply
copying half the defect from the ﬁne grid to the corresponding coarse-grid
point. The calls to resid followed by rstrct in the ﬁrst part of the
V-cycle can be replaced by a routine that loops only over the coarse grid,
ﬁlling it with half the defect.
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• Similarly, the quantity unew = uh + P vH need not be computed at red
h
mesh points, since they will immediately be redeﬁned in the subsequent
Gauss-Seidel sweep. This means that addint need only loop over black
points.
• You can speed up relax in several ways. First, you can have a special
form when the initial guess is zero, and omit the routine fill0. Next, you
can store h2 fh on the various grids and save a multiplication. Finally, it
is possible to save an addition in the Gauss-Seidel formula by rewriting
it with intermediate variables.
• On typical problems, mglin with ncycle = 1 will return a solution with
the iteration error bigger than the truncation error for the given size of h.
To knock the error down to the size of the truncation error, you have to
set ncycle = 2 or, more cheaply, npre = 2. A more efﬁcient way turns
out to be to use a higher-order P in (19.6.20) than the linear interpolation
used in the V-cycle.
Implementing all the above features typically gives up to a factor of two
improvement in execution time and is certainly worthwhile in a production code.
Nonlinear Multigrid: The FAS Algorithm
Now turn to solving a nonlinear elliptic equation, which we write symbolically as
L(u) = 0 (19.6.21)
Any explicit source term has been moved to the left-hand side. Suppose equation (19.6.21)
is suitably discretized:
Lh (uh ) = 0 (19.6.22)
We will see below that in the multigrid algorithm we will have to consider equations where a
nonzero right-hand side is generated during the course of the solution:
Lh (uh ) = fh (19.6.23)
One way of solving nonlinear problems with multigrid is to use Newton’s method, which
produces linear equations for the correction term at each iteration. We can then use linear
multigrid to solve these equations. A great strength of the multigrid idea, however, is that it
can be applied directly to nonlinear problems. All we need is a suitable nonlinear relaxation
method to smooth the errors, plus a procedure for approximating corrections on coarser grids.
This direct approach is Brandt’s Full Approximation Storage Algorithm (FAS). No nonlinear
equations need be solved, except perhaps on the coarsest grid.
To develop the nonlinear algorithm, suppose we have a relaxation procedure that can
smooth the residual vector as we did in the linear case. Then we can seek a smooth correction
vh to solve (19.6.23):
Lh (uh + vh ) = fh (19.6.24)
To ﬁnd vh, note that
Lh (uh + vh ) − Lh (uh ) = fh − Lh (uh )
(19.6.25)
= −dh

19.6 Multigrid Methods for Boundary Value Problems 883
The right-hand side is smooth after a few nonlinear relaxation sweeps. Thus we can transfer
the left-hand side to a coarse grid:
LH (uH ) − LH (Ruh ) = −Rdh (19.6.26)
that is, we solve
LH (uH ) = LH (Ruh ) − Rdh (19.6.27)
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on the coarse grid. (This is how nonzero right-hand sides appear.) Suppose the approximate
solution is uH . Then the coarse-grid correction is
vH = uH − Ruh (19.6.28)
and
unew = uh + P(uH − Ruh )
h (19.6.29)
Note that PR = 1 in general, so unew = P uH . This is a key point: In equation (19.6.29) the
h
interpolation error comes only from the correction, not from the full solution uH .
Equation (19.6.27) shows that one is solving for the full approximation uH , not just the
error as in the linear algorithm. This is the origin of the name FAS.
The FAS multigrid algorithm thus looks very similar to the linear multigrid algorithm.
The only differences are that both the defect dh and the relaxed approximation uh have to be
restricted to the coarse grid, where now it is equation (19.6.27) that is solved by recursive
invocation of the algorithm. However, instead of implementing the algorithm this way, we
will ﬁrst describe the so-called dual viewpoint, which leads to a powerful alternative way
of looking at the multigrid idea.
The dual viewpoint considers the local truncation error, deﬁned as
τ ≡ Lh (u) − fh (19.6.30)
where u is the exact solution of the oiginal continuum equation. If we rewrite this as
Lh (u) = fh + τ (19.6.31)
we see that τ can be regarded as the correction to fh so that the solution of the ﬁne-grid
equation will be the exact solution u.
Now consider the relative truncation error τh , which is deﬁned on the H-grid relative
to the h-grid:
τh ≡ LH (Ruh) − RLh (uh ) (19.6.32)
Since Lh (uh ) = fh , this can be rewritten as
LH (uH ) = fH + τh (19.6.33)
In other words, we can think of τh as the correction to fH that makes the solution of the
coarse-grid equation equal to the ﬁne-grid solution. Of course we cannot compute τh , but we
do have an approximation to it from using uh in equation (19.6.32):
τh τh ≡ LH (Ruh ) − RLh (uh ) (19.6.34)
Replacing τh by τh in equation (19.6.33) gives
LH (uH ) = LH (Ruh ) − Rdh (19.6.35)
which is just the coarse-grid equation (19.6.27)!
Thus we see that there are two complementary viewpoints for the relation between
coarse and ﬁne grids:
• Coarse grids are used to accelerate the convergence of the smooth components
of the ﬁne-grid residuals.

884 Chapter 19. Partial Differential Equations
• Fine grids are used to compute correction terms to the coarse-grid equations,
yielding ﬁne-grid accuracy on the coarse grids.
One beneﬁt of this new viewpoint is that it allows us to derive a natural stopping criterion
for a multigrid iteration. Normally the criterion would be
dh ≤ (19.6.36)
and the question is how to choose . There is clearly no beneﬁt in iterating beyond the
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point when the remaining error is dominated by the local truncation error τ . The computable
quantity is τh . What is the relation between τ and τh ? For the typical case of a second-order
accurate differencing scheme,
τ = Lh (u) − Lh (uh ) = h2 τ2 (x, y) + · · · (19.6.37)
Assume the solution satisﬁes uh = u + h2 u2 (x, y) + · · · . Then, assuming R is of high
enough order that we can neglect its effect, equation (19.6.32) gives
τh LH (u + h2 u2 ) − Lh (u + h2 u2 )
= LH (u) − Lh (u) + h2 [LH (u2 ) − Lh (u2 )] + · · · (19.6.38)
= (H 2 − h2 )τ2 + O(h4 )
For the usual case of H = 2h we therefore have
1 1
τ τ
3 h
τ
3 h
(19.6.39)
The stopping criterion is thus equation (19.6.36) with
= α τh , α∼ 1
3 (19.6.40)
We have one remaining task before implementing our nonlinear multigrid algorithm:
choosing a nonlinear relaxation scheme. Once again, your ﬁrst choice should probably be
the nonlinear Gauss-Seidel scheme. If the discretized equation (19.6.23) is written with
some choice of ordering as
Li (u1 , . . . , uN ) = fi , i = 1, . . . , N (19.6.41)
then the nonlinear Gauss-Seidel schemes solves
Li (u1 , . . . , ui−1 , unew , ui+1 , . . . , uN ) = fi
i (19.6.42)
for unew . As usual new u’s replace old u’s as soon as they have been computed. Often equation
i
(19.6.42) is linear in unew , since the nonlinear terms are discretized by means of its neighbors.
i
If this is not the case, we replace equation (19.6.42) by one step of a Newton iteration:
Li (uold ) − fi
unew = uold −
i i
i
(19.6.43)
∂Li (uold )/∂ui
i
For example, consider the simple nonlinear equation
2
u + u2 = ρ (19.6.44)
In two-dimensional notation, we have
L(ui,j ) = (ui+1,j + ui−1,j + ui,j+1 + ui,j−1 − 4ui,j )/h2 + u2 − ρi,j = 0 (19.6.45)
i,j
Since
∂L
= −4/h2 + 2ui,j (19.6.46)
∂ui,j
the Newton Gauss-Seidel iteration is
L(ui,j )
unew = ui,j −
i,j (19.6.47)
−4/h2 + 2ui,j
Here is a routine mgfas that solves equation (19.6.44) using the Full Multigrid Algorithm
and the FAS scheme. Restriction and prolongation are done as in mglin. We have included
the convergence test based on equation (19.6.40). A successful multigrid solution of a problem
should aim to satisfy this condition with the maximum number of V-cycles, maxcyc, equal to
1 or 2. The routine mgfas uses the same functions copy, interp, and rstrct as mglin.